Scale Drawings and Scale Factor

Transcription

1 Scale Drawings and Scale Factor Fleas are some of the animal kingdom's most amazing athletes. Though they are on average only 16 1 inch long, they can leap up to seven inches vertically and thirteen inches horizontally. This helps them attach themselves to warm-blooded hosts, such as people, dogs, or cats. 8.1 Bigger and Smaller Scale Drawings, Scale Models, and Scale Factors Say Cheese! Applications of Ratio No GPS? Better Get the Map Out! Exploring Scale Drawings Houses for Our Feathered Friends! Creating Blueprints

2 Chapter 8 Overview This chapter explores scale drawings, scale models, and scale factors through real-world problems. Lessons TEKS Pacing Highlights Models Worked Examples Peer Analysis Talk the Talk Technology 8.1 Scale Drawings, Scale Models, and Scale Factors 7.5.A 7.5.C 1 This lesson presents real-world situations to explore scale drawings and scale factor through measurement. Questions ask students to calculate actual measurements using a given scale factor. X 8.2 Applications of Ratio 7.5.A 7.5.C 1 This lesson provides the aspect ratios for different real-world objects. Questions ask students to use aspect ratios to determine other measurements. X 8.3 Exploring Scale Drawings 7.5.A 7.5.C 2 This lesson develops an understanding of a scale on a map or drawing. Questions ask students to design a courtyard for the school using a given scale. X X 8.4 Creating Blueprints 7.5.C 1 This lesson provides two blueprints and asks students to label the parts of each blueprint with the appropriate measures. X 431A Chapter 8 Scale Drawings and Scale Factor

3 Skills Practice Correlation for Chapter 8 Lessons Problem Set Objective(s) Scale Drawings, Scale Models, and Scale Factors Applications of Ratio Exploring Scale Drawings Creating Blueprints Vocabulary 1 6 Enlarge or shrink drawings using grids 7 12 Use scale factors to answer questions Vocabulary 1 6 Calculate unknown dimensions of rectangles 7-12 Calculate unknown dimensions given the aspect ratio Calculate aspect ratio of rectangles Use given information to answer aspect ratio questions Vocabulary 1 6 Write scale ratios to represent sentences 7 12 Determine distances on a map using a scale Choose scales as described Determine whether scales are larger or smaller than original objects Use given information to answer scale questions 1 4 Label measurements of construction pieces for items to be built 5 8 Draw and label blueprints for items Chapter 8 Scale Drawings and Scale Factor 431B

4 432 Chapter 8 Scale Drawings and Scale Factor

5 Bigger and Smaller Scale Drawings, Scale Models, and Scale Factors Learning Goals In this lesson, you will: Use scale models to calculate measurements. Use scale factors to enlarge and shrink models. Key Term scale factor Essential Ideas Scale drawings are used to display very large and very small objects. One method used for enlarging or reducing a drawing is to use a grid. Scale models are used to show a three-dimensional dilation. Texas Essential Knowledge and Skills for Mathematics Grade 7 (5) Proportionality. The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships. The student is expected to: (A) generalize the critical attributes of similarity, including ratios within and between similar shapes (C) solve mathematical and real-world problems involving similar shape and scale drawings Materials Ruler 8.1 Scale Drawings, Scale Models, and Scale Factors 433A

6 Overview Students use scale models to calculate measurements and use dilations to enlarge and reduce the size of models. Students encounter real world situations involving maps, blueprints, atoms, grid drawings, model airplanes, model barns, and dollhouses. In each of these situations, they will enlarge or reduce the size of objects and calculate relevant measurements. 433B Chapter 8 Scale Drawings and Scale Factor

7 HI Warm Up A WA MT OR ID WY NV UT CO CA AZ NM AK VT NH ME ND MN MA SD WI NY MI RI CT NE IA PA NJ OH IL IN DE WV KS VA MD MO KY DC NC TN OK AR SC MS AL GA LA TX FL B WA OR NV CA AK ID AZ UT MT WY CO NM VT NH ND MN SD WI NY MI NE IA PA OH IL IN WV VA KS MO KY TN NC OK AR SC MS AL GA LA TX FL ME MA RI CT NJ DE MD DC HI C Two maps of the United States are exact copies but different sizes. Use point A as the center of dilation and measure the distance between points A and B, and points A and C to calculate the scale factor from the smaller map to the larger map. The scale factor is Scale Drawings, Scale Models, and Scale Factors 433C

8 433D Chapter 8 Scale Drawings and Scale Factor

9 Bigger and Smaller Scale Drawings, Scale Models, and Scale Factors Learning Goals In this lesson, you will: Use scale models to calculate measurements. Use scale factors to enlarge and shrink models. Key Term scale factor Some professional basketball players can jump really high to dunk the ball on a 10-foot-tall goal. But those athletes have got nothing on fleas. These parasitic insects, which spend their time trying to suck the blood from other animals, can jump as high as seven inches. That doesn t sound impressive unless you know that a flea is only about 1 16 inch long, which means that a flea can jump more than 100 times its own length! If you could jump like a flea, how high could you jump? What tall buildings could you leap in a single bound? 8.1 Scale Drawings, Scale Models, and Scale Factors 433

10 Problem 1 Two drawings of a sailboat are shown that are the exact copies but different sizes. Students measure the dimensions of the mainsail in both the original and enlarged pictures to calculate the scale factor. In the next question, students will calculate the scale factor from blueprints of a house. Finally, students use a grid to enlarge a drawing of a sailboat, given the image of a smaller sailboat. Problem 1 Scale Drawings 1. Emma enrolled in a sailing class. This diagram of a sailboat is on the first page of her text. Jib Mainsail Hull Jib Sheet Centerboard Rudder She decided to enlarge the diagram on a separate piece of paper as shown. Materials Ruler Grouping Have students complete Questions 1 through 4 with a partner. Then share the responses as a class. Jib Mainsail Share Phase, Question 1 Which feature(s) of the sailboat resembled a triangle? Which feature(s) of the sailboat resembled a quadrilateral? Which feature(s) of the sailboat resembled a pentagon? Jib Sheet Centerboard Hull Rudder 434 Chapter 8 Scale Drawings and Scale Factor

11 Explain to students that a scale factor is the same as a ratio, which they have learned previously. The scale factor, or ratio, is the number you multiply by when enlarging or shrinking an object. Help students to identify scale factors which show an increase in size or enlargement and those which represent a decrease or shrinking in size. For example, scale factors greater than 1 represent an enlargement, and scale factors less than 1 represent a decrease in size. 2. Determine the geometric shape that best describes each part of the sailboat. Mainsail Triangle Hull Pentagon Centerboard Trapezoid Jib Triangle Rudder Rectangle 3. Use a centimeter ruler to measure the dimensions of the Mainsail in the text and the Mainsail in Emma s enlargement of the diagram. 4. The ratio of side lengths in the enlargement to those of the original figure is called the scale factor. Determine the scale factor Emma used to create the enlargement of the diagram. A blueprint is an example of a scale drawing that represents a larger structure. The blueprint shown will be used for the construction of a new house. HERS PORCH MASTER BED LIN. BED BREAKFAST 9-0/11 Trey Clg. Hgt HIS 9-0Clg. Hgt Clg. Hgt M. BATH 9-0 Clg. Hgt B-2 BED Clg. Hgt FAMILY ROOM /10/11 TREY Clg. Hgt GALLERY BED 4 / STUDY KITCHEN Clg. Hgt DINING ROOM FOYER 10-0 Clg. Hgt PORCH UTILITY STORAGE 2 CAR GARAGE Share Phase, Questions 2 through 4 How did you determine the scale factor of the two sailboats? How did you determine the scale factor was 2? If the scale factor is equal to 2, what does that mean about the original sailboat and the enlarged image of sailboat, or the image of dilation? Grouping Have students complete Question 5 with a partner. Then share the responses as a class. 5. Use a centimeter ruler to determine the scale factor used to create the blueprint. Answers will vary. The scale factor for the blueprint is approximately Do I have to measure everything? STORAGE Share Phase, Question 5 How did you determine the scale factor for the blueprint? Could there be more than one correct answer? Why or why not? 8.1 Scale Drawings, Scale Models, and Scale Factors 435

12 Grouping Ask a student to read the information aloud. Discuss the information as a class. Discuss Phase Using what you know about drawings, sketches, and constructions, do you think a drawing can accurately display a scale drawing of a tiny object? Can you think of other objects that are 10,000 times smaller than an original figure or object? Scale drawings are also used to display small objects. The illustration shown is an artist s drawing of an oxygen atom. It shows eight electrons orbiting a nucleus that contains eight protons (dark spheres) and eight neutrons (light spheres). If the drawing were to scale, the nucleus would be invisible, 10,000 times smaller than it is currently drawn. A more sophisticated depiction of the electrons would show them as pulsating, three-dimensional wavelike clouds rather than little orbiting bullets. One method for enlarging or shrinking a drawing is to use a grid. The drawing of the sailboat that follows has been made on a grid. Another grid with larger cells is drawn. The idea is to copy each portion of the drawing that is in each square of the original grid into the corresponding square of the new grid. Re-read the previous two paragraphs and think about the illustration of the atom. How would you use a grid to draw the atom to scale as the artist has done? As you think about using a grid to enlarge or shrink a drawing, use precise mathematical language and circle those terms on the page. 436 Chapter 8 Scale Drawings and Scale Factor

13 Grouping Have students complete Question 6 with a partner. Then share the responses as a class. 6. Use this method to enlarge the drawing. Share Phase, Question 6 How do you know where to begin when you used a grid to enlarge the drawing of the sailboat? Can you begin at more than one location? Are there any advantages to using a grid to enlarge or reduce an image? What are they? Jib Sheet Centerboard Mainsail Hull Rudder Mainsail Jib Sheet Centerboard Hull Rudder 8.1 Scale Drawings, Scale Models, and Scale Factors 437

14 Problem 2 Scale models are used to show a three-dimensional dilation. A model plane, a model of a barn, and a dollhouse are used as the context for this problem. The model plane has a scale of 1, the model barn used 100 a scale of 1 to 48, and the dollhouse has a scale of 1:12. Students will use the information given to determine the measurements of the original objects in the problem situations. Problem 2 Scale Models Scale models are also used for three-dimensional models. 1. A model of a C130 airplane has a scale of a. If the model plane is one foot long, how long is the actual plane? The plane is 100 feet long. b. If the model s wingspan is 16 inches, how long is the actual wingspan? The actual wingspan is 133 feet 4 inches. Grouping Have students complete Questions 1 and 2 with a partner. Then share the responses as a class. c. If the width of each of the model s propellers is 1.62 inches, how wide is an actual propeller? The width of an actual propeller is 13 feet 6 inches. Share Phase, Question 1 How did you calculate the length of the actual plane? How many inches are in one foot? How do you change 1600 inches into feet? How many feet are 1600 inches? How did you calculate the actual wingspan of the plane? How do you change 162 inches into feet? How did you calculate the width of the propeller of the model plane? How many feet are in 162 inches? d. If the width of the actual tail is 52 feet 8 inches, what is the width of the tail in the model? The width of the model s tail is 6.32 inches. e. If the height of the actual tail is 38 feet 5 inches, what is the height of the tail in the model? The height of the tail in the model is 4.61 inches. How did you calculate the width of the tail of the model plane? How many inches are in 52 feet 8 inches? How did you calculate the height of the tail of the model plane? How many inches are in 38 feet 5 inches? 438 Chapter 8 Scale Drawings and Scale Factor

15 Share Phase, Question 2 How did you calculate the height of the actual barn door? How do you write two and one quarter inches using decimals? How do you change 1296 inches into feet? How many feet are in 1296 inches? How did you calculate the actual height of the silo? How many inches are in 80 feet? How many inches are in 50 feet? How many inches are in 60 feet? How did you calculate the dimensions of the model of the barn? 2. This model of a barn has been constructed using a scale of 1 to 48. a. If the model s barn door is two and one quarter inches high, how high is the actual barn door? The actual barn door is 9 feet, or 108 inches, high. b. If the model s silo is 18 inches high, how high is the actual silo? The actual silo is 72 feet high. c. The actual barn is 80 feet wide, 50 feet deep, and 60 feet to the roof. What are the dimensions of the model? The model is 20 inches wide, 12.5 inches deep, and 15 inches to the roof. 3. a. Suppose a dollhouse is built using a scale of 1 : 12. The actual house has 10 foot ceilings in all the rooms. How high are the ceilings in the dollhouse? The ceilings in the dollhouse are 10 inches high. b. The porch on the dollhouse is 6 inches high. How high is the actual porch of the house? The actual porch is 6 feet high. Be prepared to share your solutions and methods. 8.1 Scale Drawings, Scale Models, and Scale Factors 439

16 Follow Up Assignment Use the Assignment for Lesson 8.1 in the Student Assignments book. See the Teacher s Resources and Assessments book for answers. Skills Practice Refer to the Skills Practice worksheet for Lesson 8.1 in the Student Assignments book for additional resources. See the Teacher s Resources and Assessments book for answers. Assessment See the Assessments provided in the Teacher s Resources and Assessments book for Chapter 8. Check for Students Understanding 1. The majority of die cast model cars made by a particular company are 1:18, what does this mean? This means the model car is 18 times smaller than the actual car. 2. The image shown illustrates the comparative average sizes of some of the more popular scales of die cast model cars available. Explain what all of the ratios mean in this chart. The smallest model car on the chart is approximately 7 cm in length, or 64 times smaller than the actual car. The company also makes model cars that are approximately 10 cm in length (43 times smaller than the actual car), approximately 18 cm in length (24 times smaller than the actual car), and approximately 24 cm in length (18 times smaller than the actual car). 440 Chapter 8 Scale Drawings and Scale Factor

17 3. The message on the bottom of the chart says Illustrating approximate representative sizes Not to scale, what does this mean? It means the images of the model cars are not exact, but approximate. Not drawn to scale means the dimensions of the image of the model cars may slightly off from the actual size of the model car. 8.1 Scale Drawings, Scale Models, and Scale Factors 440A

18 440B Chapter 8 Scale Drawings and Scale Factor

19 Say Cheese! Applications of Ratio Learning Goal In this lesson, you will: Work with applications of similarity and scale factor. Key Term aspect ratio Essential Ideas An aspect ratio of an image is the ratio of its width to its height. Aspect ratios are written as two numbers separated by a colon (width: height). Texas Essential Knowledge and Skills for Mathematics Grade 7 (5) Proportionality. The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships. The student is expected to: (A) generalize the critical attributes of similarity, including ratios within and between similar shapes (C) solve mathematical and real-world problems involving similar shape and scale drawings 8.2 Applications of Ratio 441A

20 Overview Students explore the aspect ratios associated with similar objects related to photographs, televisions, flags, and Lego structures. The worlds of Lilliput and Brobdingnag of Gulliver s Travels are used to compare the size of objects found in our world. Scale factors are connected to aspect ratios. Students will use aspect ratios and scale factors to determine the actual size of a car compared to a toy car, and the size of an actual train compared to a model train. 441B Chapter 8 Scale Drawings and Scale Factor

21 Warm Up 1. If the scale factor of the small heart to the big heart is 1:5, what does this mean? It means the small heart is 1 the size of the large heart, or the large heart is five times the size 5 of the small heart. 2. If the scale factor of the large star to the small star is 4:1, what does this mean? It means the large star is four times the size of the small star, or the small star is 1 the size of 4 the large star. 8.2 Applications of Ratio 441C

22 441D Chapter 8 Scale Drawings and Scale Factor

23 Say Cheese! Applications of Ratio Learning Goal In this lesson, you will Work with applications of similarity and scale factor. Key Term aspect ratio Up until the 1920s, movies did not have any sound with them. These silent films had what were known as intertitles to show dialogue and to tell the story being shown. These movies were far from silent, however. They were often played in a theater and live music was played to the action of the movie. Have you ever seen a silent film? 8.2 Applications of Ratio 441

24 Problem 1 In the problem situation, all of the photographs sold by a particular photography company are in the shape of similar rectangles. The largest size photo measures 12 inches in width by 16 inches in height. Students will determine the dimensions of other possible photographs. Problem 1 School Photos When Timmons Photo Company prints photo packages, they include several sizes of photos that are all mathematically similar. The largest size is 12 in in. This is read as 12 inches by 16 inches. The first 16 in. measure is the width of the photo, and the second measure is the height of the photo. 12 in. 1. Determine the width of a mathematically similar photo that has a height of 8 inches. Grouping Have students complete Questions 1 and 2 with a partner. Then share the responses as a class. Share Phase, Question 1 How is it possible that both students used different ratios but still arrived at the same answer? Do you prefer Fran s method or Joe s method, and why? Fran To determine the unknown width of the smaller photo, I wrote ratios using the measurements from within each photo. The ratio on the left contains the measurements of the large photo. The ratio on the right contains the measurements of the smaller photo. width of largest photo height of largest photo width of smaller photo = height of smaller photo 12 inches 16 inches = x inches 8 inches (12)(8) = (16)(x) 96 = 16x 6 = x I calculated that the width of the smaller photo is 6 inches. Joe To determine the unknown width of the smaller photo, I wrote ratios using the measurements between the two photos. The ratio on the left contains the width measurements of both photos. The ratio on the right contains the height measurements of both photos. width of larger photo width of smaller photo height of larger photo = height of smaller photo 12 inches 16 inches = x inches 8 inches (12)(8) = (x)(16) 96 = 16x 6 = x I calculated that the width of the smaller photo is 6 inches. 442 Chapter 8 Scale Drawings and Scale Factor

25 a. What is similar about the two solution methods? What is different about the two solution methods? A similarity of the two methods is that both calculated the correct answer. A difference between the two methods is that Fran wrote ratios that contained measurements from within each photo, but Joe wrote ratios that contained measurements between the two photos. b. Fran originally tried using the following proportion to calculate the width of the smaller photo, then determined it was incorrect. width of largest photo height of largest photo 5 height of smaller photo width of smaller photo What is incorrect with the proportion? The problem is that the measurements in the two ratios are not aligned. To maintain equality in the proportion, write the width measurement in the numerator of both ratios and the length measurement in the denominator of both ratios. Another option is to write the height measurement in the numerator of both ratios and the width measurement in the denominator of both ratios. width of largest photo Correct Proportions: height of largest photo height of largest photo width of largest photo 5 width of smaller photo height of smaller photo OR height of smaller photo 5 width of smaller photo Share Phase, Question 2 If the photograph is 2 inches wide, how do you determine the height? If the photograph is 8 inches in height, how do you determine the width? If the photograph is 3 inches wide, how do you determine the height? If the photograph is 2 inches in height, how do you determine the width? If the photograph is 4 inches wide, how do you determine the height? If the photograph is 3.5 inches in height, how do you determine the width? 2. Determine the other possible photo sizes that are mathematically similar. a. 2 in in., or in. b. 3 in. 3 4 in. c. 1.5 in. 3 2 in. d. 4 in. 3 e. 2.4 in in in., or in. 8.2 Applications of Ratio 443

26 Problem 2 The term aspect ratio is introduced within the context of screen sizes for movie theaters and televisions. Students should notice that aspect ratio is written similarly to scale with two numbers separated by a colon. They will then need to ensure the differences between aspect ratio and scale factor. The Check for Understanding exercise for this lesson was written to emphasize the difference. Students will complete tables that list possible screen sizes for high definition televisions and Panavision, given the standard aspect ratios of each. Problem 2 Aspect Ratios An aspect ratio of an image is the ratio of its width to its height. Aspect ratios are used to determine the screen sizes for movie screens and televisions. Aspect ratios are written as two numbers separated by a colon (width : height). height width 1. Before 1950, the aspect ratio of all motion pictures and standard definition televisions was 1.33 : 1. This meant that the screen was 1.33 times as wide as it was tall. a. Scale this ratio up to a ratio using only whole numbers. 4 : 3 b. What did you use for your scale factor? Explain how you determined what scale factor to use. I used 3 as my scale factor. I knew that I had to multiply 1.33, or 4, by 3 to 3 eliminate the fraction or decimal and get a whole number. Grouping Have students complete Questions 1 through 4 with a partner. Then share the responses as a class. Share Phase, Questions 1 and 2 How did you scale up the ratio describing standard definition televisions? What does widescreen mean? How can you tell if you have a widescreen television? 2. After 1950, the movie industry wanted to create a different image than what was seen on television, so it adopted the widescreen ratios of 1.85 : 1, which was called the Academy Flat, and 2.35 : 1, which was called Panavision. Explain why these ratios are called widescreen ratios. The original aspect ratio was 1.33 : 1. The width was 1.33 inches for every 1 inch of height. The widescreen ratios had greater widths, 1.85 inches and 2.35 inches, respectively, for the same height of 1 inch. 444 Chapter 8 Scale Drawings and Scale Factor

27 Share Phase, Questions 3 and 4 How can you tell if you have a high definition television with an incorrect aspect ratio? If the width of a high definition television is 8 inches, how do you determine the height, with respect to the suggested aspect ratio? If the height of a high definition television is 18 inches, how do you determine the width, with respect to the suggested aspect ratio? If the height of a Panavision screen is 1 foot, how do you determine the width, with respect to the suggested aspect ratio? If the width of a Panavision screen is 11.5 feet, how do you determine the height, with respect to the suggested aspect ratio? 3. High definition televisions, or HDTVs, use an aspect ratio of 1.78 : 1. Written as a ratio using whole numbers, the HDTV aspect ratio is 16 : 9. Complete the table to show which similar television screen sizes are appropriate for showing TV shows and movies in high definition. Width HDTV Sizes Height 8 inches 4.5 inches 32 inches 18 inches 48 inches 27 inches 5 ft 4 in. 3 feet 8 feet 4.5 feet 4. Complete the table to show which similar television screen sizes are appropriate for to show movies made in Panavision. Width Panavision Sizes Height 2.35 feet 1 foot 14.1 feet 6 feet 28.2 feet 12 feet 4.5 feet is 54 inches feet 5 feet 23.5 feet 10 feet 47 feet 20 feet 70.5 feet 30 feet 8.2 Applications of Ratio 445

28 Problem 3 Each country can be associated with a unique flag and each flag has a specified height to length ratio. Students analyze a given table that groups several countries by certain flag size ratios of height to length. They will determine the group in which each flag belongs based on the aspect ratios given in the table. Problem 3 Flags of the World Each country of the world has a flag that is designed to a specific ratio of height : length. All the flags of a particular country must be proportioned in the same ratio. length height Grouping Have students complete Questions 1 through 3 with a partner. Then share the responses as a class. Hmm... Which group does a flag with has a height of 12 feet and a length of 20 feet belong to? The table shown lists some countries and the height : length ratio of their flags. Group A Bermuda Canada Ethiopia Jamaica Countries Libya New Zealand Nigeria Ratio height : length 1 : 2 Group B Liberia United States 10 : 19 Group C China Congo Egypt France Greece India Italy Japan Kenya Russia South Africa Spain 2 : 3 Group D Iran Mexico 4 : 7 Group E England Germany Haiti Nicaragua Scotland Wales 3 : 5 Group F Switzerland Vatican City 1 : Chapter 8 Scale Drawings and Scale Factor

29 Share Phase, Questions 1 through 3 If the flag is 2 feet in height and 4 feet in width, what is the aspect ratio? If the flag is 10 feet in height and 15 feet in width, what is the aspect ratio? If the flag is 20 feet in height and 20 feet in width, what is the aspect ratio? If the flag is 12 feet in height and 21 feet in width, what is the aspect ratio? If the flag is 5 feet in height and 9.5 feet in width, what is the aspect ratio? If the flag is 1.5 feet in height and 2.5 feet in width, what is the aspect ratio? 1. The sizes of flags are given in terms of height 3 length for each. State which group (A through F) each flag must belong to based on its ratio of height : length. a. 2 feet 3 4 feet 2 feet : 4 feet 5 1 : 2 Group A b. 10 feet 3 15 feet 10 feet : 15 feet 5 2 : 3 Group C c. 20 feet 3 20 feet 20 feet : 20 feet 5 1 : 1 Group F d. 12 feet 3 21 feet 12 feet : 21 feet 5 4 : 7 Group D e. 5 feet feet 5 feet : 9.5 feet 5 10 : 19 Group B f. 1.5 feet feet 1.5 feet : 2.5 feet 5 3 : 5 Group E 2. Which group of countries has square flags? Group F. For square flags, the ratio of the height to the length is 1 : Which groups of countries have flags which are slightly different from 1 : 2? Group B, Group D, Group E 8.2 Applications of Ratio 447

30 Problem 4 Lego s are used to build a replica of the Empire State building. Four different aspect ratios were used in the construction of the building to accentuate the height of the structure. Students approximate the actual height of the building using the information provided in the problem situation. They will then complete a table of famous buildings by calculating either the height of the actual building or the height of a scale model. Problem 4 Legoland Legoland, California, has an area called Miriland, USA with all the famous U.S. buildings built to a 1 : 20 or 1 : 40 scale. One exception is the Empire State Building. The model of the Empire State Building is built using four different scales. The ground floors are built at a 1 : 20 scale to match the size of the model people on the street. The main body of the building is built at a 1 : 40 scale. It then changes to a 1 : 60 scale closer to the top of the model, and the very top tower is built at a 1 : 80 scale. The different scales at the higher levels of the model trick the eye into thinking that the building is much taller than it is. If you were to build a model of the Empire State Building using a 1 : 20 scale for the entire model, it would be over 62 feet tall versus the Legoland version, which is 20 feet tall! 1. Approximately how tall is the Empire State Building? Use the fact that a 1 : 20 scale model would be over 62 feet tall. Show and explain your work. The Empire State Building must be approximately 1,240 feet tall. I calculated the height by multiplying the 62 foot height of the model by 20. While students work in pairs to determine the heights of the actual buildings and the models, ask them to share their processes. Challenge students to work cooperatively to determine alternative methods for determining each height. 2. Complete the table to represent the heights of actual buildings and the heights of their models at a 1 : 20 scale. Name of Building Washington Monument Washington, D.C. U.S. Capitol Building Washington, D.C. Willis Tower (formerly the Sears Tower) Chicago, Illinois Transamerica Pyramid San Francisco, CA Height of the Actual Building Height of the Scale Model at a 1 : 20 Scale feet feet 88 meters 4.4 meters 1451 feet feet 850 feet 42.5 feet Grouping Have students complete Questions 1 and 2 with a partner. Then share the responses as a class. 191 Peachtree Tower Atlanta, GA Modis Tower Jacksonville, FL 265 m m m m Share Phase, Questions 1 and 2 How did you determine the height of the scale model of the Washington Monument? How did you determine the actual height of the U.S. Capitol Building? How did you determine the height of the scale model of the Willis Tower? 448 Chapter 8 Scale Drawings and Scale Factor

31 Problem 5 This problem uses the imaginary worlds of Lilliput and Brobdingnag from Jonathan Swift s Gulliver s Travel s. Lilliput is the land of the tiny people ( 1 the size of Lemuel), and 12 Brobdingnag (12 times the size of Lemuel) is the land of the giants. Using this information, students complete a table in which they compute the measurements of several objects in the real world, in the Lilliput world, and in the Brobdingnag world. Grouping Have students complete Question 1 with a partner. Then share the responses as a class. Problem 5 Gulliver s Travels Maybe you have read or seen Gulliver s Travels, written by Jonathan Swift and published in In the story, Lemuel Gulliver visits two lands in his travels: Lilliput, the land of tiny people, and Brobdingnag, the land of the giants. The Lilliputians are 1 of Lemuel s size, 12 and the Brobdingnagians are 12 times his size. 1. Complete the measurements in the table to compare your world, which is the same as Lemuel s, with the worlds of the Lilliputians and the Brobdingnagians. Answers will vary. Your World Lilliput World Brobdingnag World a. Pencil Length 6 in. 0.5 in. 72 in., or 6 ft b. Your Height 56 in in. 672 in., or 56 ft c. Math Book Length and Width 8 in in in in. 96 in in., or 8 ft 3 11 ft d. Your Foot Length 7 in in. 84 in., or 7 ft Share Phase, Question 1 How did you determine the length of the objects in our world? How did you determine the length of the objects in the Lilliput world? How did you determine the length of the objects in the Brobdingnag world? How did you determine your height in our world? How did you determine your height in the Lilliput world? How did you determine your height in the Brobdingnag world? e. Paper Clip Length 1.25 in in. 15 in. f. Postage Stamp Length and Width 1 in in in in. Be sure to label your measurements. 12 in in. 8.2 Applications of Ratio 449

32 Problem 6 Statements that describe the scale factors used to create toy cars and model trains are given. Students interpret each statement with respect to the actual size of the objects. Problem 6 Models 1. The scale factor for a model car is 1 : 24. What does this mean? Grouping Have students complete Questions 1 and 2 with a partner. Then share the responses as a class. The model car is 1 the size of an actual car. An actual car is 24 times the size of 24 the model car. Share Phase, Questions 1 and 2 How do you read 1:24? Is it the same as 24:1? How is it different than 24:1? How does the size of a toy car compare to the size of an actual car? How does an actual car compare to the size of a toy car? How do you read 1:87? Is it the same as 87:1? How is it different than 87:1? How does the size of a model train compare to the size of an actual train? How does an actual train compare to the size of a toy train? 2. The scale factor for a model train is 1 : 87. What does this mean? The model train is 1 the size of an actual train. An actual train is 87 times the size 87 of the model train. Be prepared to share your solutions and methods. 450 Chapter 8 Scale Drawings and Scale Factor

33 Follow Up Assignment Use the Assignment for Lesson 8.2 in the Student Assignments book. See the Teacher s Resources and Assessments book for answers. Skills Practice Refer to the Skills Practice worksheet for Lesson 8.2 in the Student Assignments book for additional resources. See the Teacher s Resources and Assessments book for answers. Assessment See the Assessments provided in the Teacher s Resources and Assessments book for Chapter 8. Check for Students Understanding 1. What is an aspect ratio? An aspect ratio is the ratio of an object s width to the object s height. 2. What is a scale factor? A scale factor is the ratios of lengths of the corresponding sides of two figures. 3. If the aspect ratio of the rectangles used in a company s logo is 3:5, what does this mean? It means for every 3 units in width, the rectangle is 5 units in height. 4. Draw three possible rectangles that could be in the company s logo and label the dimensions. Answers will vary Describe the scale factors used when you created the three rectangles in the last question. Answers will vary. The scale factor between the small rectangle and the medium sized rectangle is 1:2. The scale factor between the small rectangle and the large rectangle is 1:3 8.2 Applications of Ratio 450A

34 450B Chapter 8 Scale Drawings and Scale Factor

35 No GPS? Better Get the Map Out! Exploring Scale Drawings Learning Goals In this lesson, you will: Work with applications of similarity and scale factor. Use scale drawings and maps. Key Term scale drawings Essential Ideas Scale drawings are representations of real objects or places that are in proportion to the real objects or places they represent. The scale is given as a ratio. The scale of a drawing 5 drawing length : actual length The scale of a map 5 map distance : actual distance Texas Essential Knowledge and Skills for Mathematics Grade 7 (5) Proportionality. The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships. The student is expected to: (A) generalize the critical attributes of similarity, including ratios within and between similar shapes (C) solve mathematical and real-world problems involving similar shape and scale drawings Materials Ruler 8.3 Exploring Scale Drawings 451A

36 Overview Students explore scale drawings. The scale of a drawing is drawing length : actual length and the scale of a map is map distance : actual distance. Students describe the meaning of several different scales. They will analyze a partial city map of Washington, D.C. and approximate distances between different locations within the city. Then students analyze a map of the United States and will approximate distances between cities. In the next problem, students determine which scale will produce the largest and smallest drawing of an object when different units of measure are given. A popular movie sets the stage for a series of questions for students to determine the actual size of the object or the size of the model representing the object. Students conclude the lesson by creating a blueprint for a courtyard given the scale for the blueprint. 451B Chapter 8 Scale Drawings and Scale Factor

37 Warm Up 1. These bugs are not drawn to scale. What does that mean? It means the bugs are not drawn in proportion to the actual bugs. What appears to be the smallest bug may actually be larger than other bugs. What appears to be the largest bug may actually be smaller than other bugs. 2. Is it possible that the bug on the extreme left is in real life the smallest bug in the picture? Explain your reasoning. Yes. It is possible that the bug on the extreme left is in real life the smallest bug in the picture because we have no way of knowing that the bugs are drawn to scale. 8.3 Exploring Scale Drawings 451C

38 451D Chapter 8 Scale Drawings and Scale Factor

39 No GPS? Better Get the Map Out! Exploring Scale Drawings Learning Goals In this lesson, you will: Work with applications of similarity and scale factor. Use scale drawings and maps. Key Term scale drawings What do surveyors, mapmakers, architects, engineers, and builders all have in common? All of these people use scale drawings. Scale drawings are representations of real objects or places that are in proportion to the real objects or places they represent. The scale in a scale drawing is given as a ratio. Maps and blueprints are examples of scale drawings. Why do you think scale drawings are important? 8.3 Exploring Scale Drawings 451

40 Problem 1 The term scale drawing is described as the representation of real objects or places that are in proportion to the real objects or places they represent. Students will write a sentence to describe the meaning of various drawings that have different scales. A scale is written as two numbers separated by a colon, where the first number represents the length of the drawing and the second number represents the actual length of the object. Grouping Ask a student to read the introduction to Problem 1 aloud. Discuss the worked example as a class. Problem 1 Scale Drawings The purpose of a scale drawing is to represent either a very large or very small object. The scale of a drawing might be written as: 1 cm : 4 ft Drawing Actual Length Length This scale means that every 1 centimeter of length in the drawing represents 4 feet of the length of the actual object. The scale of a map might look like this: 1 in. : 200 mi Map Actual Distance Distance This scale means that every 1 inch of distance on the map represents 200 miles of actual distance. 452 Chapter 8 Scale Drawings and Scale Factor

41 Grouping Have students complete Question 1 with a partner. Then share the responses as a class. Share Phase, Question 1 When describing the scale on a drawing, what does the first number represent? When describing the scale on a drawing, what does the second number represent? If the first number is larger than the second number, what does this mean in terms of the size of the drawing and the actual size of the object? If the second number is larger than the first number, what does this mean in terms of the size of the drawing and the actual size of the object? 1. Write a sentence to describe the meaning of each. a. A scale on a map is 1 in. : 2 ft This scale means that for every 1 inch on the map there are 2 feet of actual distance. b. A scale on a drawing is 1 cm : 4 cm This scale means that for every 1 centimeter of length on the drawing there are 4 centimeters of length on the actual object. c. A scale on a drawing is 2 in. : 1 in. This scale means that for every 2 inches on the drawing there is 1 inch of the actual object. d. A scale on a drawing is 1 cm : 1 cm. This scale means that for every 1 cm of the drawing, there is 1 centimeter of the actual object. The drawing and object are the same size. 8.3 Exploring Scale Drawings 453

42 Problem 2 A partial street map of Washington D.C. and a map key is provided. Students use the key to estimate distances between locations. Miles are used in the key as the unit of measurement and all answers are approximations. Materials Ruler Problem 2 A Map of Washington, D.C. A partial map of Washington, D.C., is provided. A scale is included on the map. This scale looks a lot like a double number line... Arlington National Cemetery Visitors Center LINCOLN MEMORIAL 1 in mi POTOMAC RIVER THE WHITE HOUSE WASHINGTON MONUMENT THOMAS JEFFERSON MEMORIAL NATIONAL MALL UNION STATION U.S. CAPITOL Bring in samples of scale drawings such as maps and blueprints for students. Have students share in small groups one of the samples, identify, and state the scale factor used. Support students in the spoken word. Grouping Have students complete Questions 1 through 5 with a partner. Then share the responses as a class. 1. Complete the table to help tourist groups plan their visits to our nation s capital. Sights White House to Lincoln Memorial Lincoln Memorial to Arlington Cemetery (Visitor Center) Arlington Cemetery (Visitor Center) to Jefferson Memorial Jefferson Memorial to Washington Monument Washington Monument to U.S. Capitol U.S. Capitol to Union Station Approximate Distance Using Roads and Paths miles miles miles 1 mile 2 miles 1 2 mile 454 Chapter 8 Scale Drawings and Scale Factor

43 Share Phase, Questions 1 through 5 What is 1 of 1 of a mile? 2 2 What is 1 of of 1 2 of a mile? What fractional part of a mile is used to approximate the distance between the White House and the Lincoln Memorial? What fractional part of a mile is used to approximate the distance between the Lincoln Memorial and the Arlington Cemetery? What fractional part of a mile is used to approximate the distance between the Arlington Cemetery and the Jefferson Memorial? What fractional part of a mile is used to approximate the distance between the Jefferson Memorial and the Washington Monument? What fractional part of a mile is used to approximate the distance between the Washington Monument and the U.S. Capitol? What fractional part of a mile is used to approximate the distance between the U.S. Capitol and the Union Station? 2. Why does it make sense to use roads and paths instead of measuring directly from one sight to the next sight? To actually travel from one sight to the next, roads and paths must be followed. The direct path may involve walking through buildings or areas that are not accessible. 3. Explain how you estimated the distances between sights. I used the edge of a sheet of paper and marked the distances as I followed a path from one sight to the next. Then, I placed the edge of the paper showing the total distance against the scale. If the distance was more than one mile, I marked one mile on the edge of the paper and went back to the beginning of the scale to estimate the additional distance. 4. Why are your answers approximate distances? My answers are approximate values because it is difficult to get an accurate distance from one sight to the next. Also, the scale is in quarters of a mile, so I could estimate only distances that fall between the quarter marks. 5. What is the total miles traveled between sights? The total distance traveled between the sights is approximately miles. 8.3 Exploring Scale Drawings 455

44 Problem 3 A map of the United States and a map key is provided. Students use the key to estimate distances between different cities. Kilometers and miles are used in the key as the units of measurement and all answers are approximations. Students will express each answer in both kilometers and miles. Problem 3 A Map of the United States A map of the United States is shown. A scale is included on the map. Seattle Why is this scale different from the one in the Washington D.C. map? Augusta Materials Ruler San Franciscoco Los Angeles Chicago Washington, D.C. Grouping Have students complete Questions 1 through 9 with a partner. Then share the responses as a class. Share Phase, Questions 1 and 2 How many kilometers are on the scale? How many miles are on the scale? How do kilometers compare to miles? Which is larger? How much larger? Will the scale on the map be helpful in determining a reasonable estimation of the distance between locations? Explain. Austin 600 km 600 mi Determine the approximate distances between the locations. State the distances in miles and kilometers. 1. Washington, D.C., to San Francisco, California The distance from Washington, D.C., to San Francisco is approximately 2700 miles or 4500 kilometers. 2. Washington, D.C., to Seattle, Washington The distance from Washington, D.C., to Seattle is approximately 2800 miles or 4500 kilometers. 456 Chapter 8 Scale Drawings and Scale Factor

45 Share Phase, Questions 3 through 7 Will the distance between locations be helpful in determining how long it will take you to travel? Is it easier to use the kilometer scale or the miles scale? Why? Is the number of kilometers between locations given on any road signs in the United States? Why or why not? 3. Washington, D.C., to your state capital Answers will vary. 4. Chicago, Illinois, to Los Angeles, California The distance from Chicago to Los Angeles is approximately 2000 miles or 3200 kilometers. 5. Augusta, Maine, to Austin, Texas The distance from Augusta to Austin is approximately 2100 miles or 3300 kilometers. 6. Which is longer, a mile or a kilometer? How can you tell? A mile is longer than a kilometer. According to the scale, 600 miles is a longer distance than 600 kilometers. 7. How many kilometers make one mile? Explain how you determined your answer. Dividing the number of kilometers by the number of miles, I always got a calculation of approximately 1.6. Therefore, 1.6 kilometers is approximately 1 mile. For example, 2220 km km 1380 mi. 1 mi 8.3 Exploring Scale Drawings 457

46 8. How many days would it take to travel from Washington, D.C., to San Francisco, California, traveling at 60 miles per hour for 8 hours per day? Show your work. A trip from Washington, D.C., to San Francisco, California, would take about 5.6 days hours days. 9. Does your response to Question 8 seem realistic? Explain your reasoning. Student explanations may vary. I think the number of days is too low. The number of miles is an underestimate because there is no direct route from Washington, D.C., to San Francisco, California. Also, if I was traveling with my family that far, we would probably want to make stops and enjoy the trip. Problem 4 Students compare scales that have different units of measure to determine which scale would produce the largest and smallest drawing of an object. They will answer questions to determine whether a scale drawing is larger or smaller than the actual figure. The movie Honey, I Shrunk The Kids is used as the context where students will calculate the sizes of the models built by the special effects teams, given the actual sizes in real life. Microscopes, architectural drawings, billboards, and statues are used as the context for students to calculate either the size of the actual object or the size of the scale drawing of the object. Problem 4 Interpreting Scales Grouping 1. Which scale would produce the largest scale drawing of an object when compared to the actual object? Explain your reasoning. 1 in. : 25 in. 1 cm : 1 m 1 in. : 1 ft If each scale is written with the same units for both terms, the scales would be 1 in. : 25 in., 1 cm : 100 cm, and 1 in. : 12 in. The largest scale would be 1 : 12 or 1 in. : 1 ft. In this case, every 1 inch represents 12 inches. This would make a larger diagram than the other ratios. Have students complete Questions 1 through 4 with a partner. Then share the responses as a class. 458 Chapter 8 Scale Drawings and Scale Factor

47 Share Phase, Questions 1 through 4 How can you compare centimeters to meters? How can you convert both to the same unit of measure? How can you compare inches to feet? How can you convert both to the same unit of measure? How can you compare millimeters to meters? How can you convert both to the same unit of measure? How can you compare centimeters to millimeters? How can you convert both to the same unit of measure? What does the numerator of a scale factor represent? What does the denominator of a scale factor represent? 2. Which scale would produce the smallest scale drawing of an object when compared to the actual object? Explain your reasoning. 1 in. : 10 in. 1 cm : 10 cm 1 mm : 1 m If each scale is written with the same units for both terms, the scales would be 1 in. : 10 in., 1 cm : 10 cm, and 1 mm : 1000 mm. The smallest scale would be 1 : 1000 or 1 mm : 1 m. In this case, every 1 millimeter represents 1000 millimeters. This would make a smaller diagram than if every unit represented 10 units. 3. The scale of a drawing is 6 cm : 1 mm. Is the scale drawing larger or smaller than the actual object or place? Explain your reasoning. The scale drawing is larger than the actual object or place. Scales are written as drawing length : actual length. In this case, the first value, the drawing length, is larger than the second value, the actual length. 4. Given a scale of 5, explain how you can tell whether the drawing is bigger or smaller 4 than the actual object. The drawing is bigger than the actual object. The top value represents a length in the drawing, and the bottom value represents a length of the actual object. So since scales are ratios, you can write them in fraction form just like any other ratio. 8.3 Exploring Scale Drawings 459

48 Grouping Have students complete Question 5 with a partner. Then share the responses as a class. Have students complete Questions 6 through 13 with a partner. Then share the responses as a class. 5. In the 1989 movie Honey I Shrunk the Kids, a professor accidentally shrinks his kids to 1 of an inch with a 4 shrink ray. The kids then get accidentally sent out to the backyard. To the tiny kids, the backyard seems to have giant ants, giant bees, and grass as tall as trees! Each ant and bee were actually these sizes in real life: Length Height Width You can write a scale as actual length : drawing length. Just remember which value is which! Share Phase, Question 5 Can you compare millimeters to inches? How? How does the length of a bee compare to the length of an ant? Which is longer? How does the height of a bee compare to the height of an ant? Which is taller? How does the width of a bee compare to the width of an ant? Which is wider? What does the scale 1:40 mean? How did you determine the length of the ant model? How did you determine the length of the bee model? How did you determine the height of the ant model? How did you determine the height of the bee model? How did you determine the width of the ant model? How did you determine the width of the bee model? Ant 12 mm 3 mm 1 mm Bee 0.5 in in in. The special effects team used a scale of 1 : 40 to create models of giant ants and bees. One unit of actual length corresponded to 40 units of length on each model. Complete the table to show the sizes of the models built by the team. Length Height Width Ant 480 mm or 48 cm 120 mm or 12 cm 40 mm or 4 cm Bee 20 in. 10 in. 10 in. 6. A microscope has a scale of 100 : 1. A microorganism appears to be 0.75 inch in length under the microscope. a. How long is the microorganism? Show your work The microorganism is actually inch long. b. A microorganism is millimeter long. How long will it appear under the microscope? Show your work The microorganism will appear to be 8.5 millimeters long under the microscope. 460 Chapter 8 Scale Drawings and Scale Factor

49 Share Phase, Questions 6 through 11 What does the microscope scale 100:1 mean? What does the microscope scale 1000:1 mean? Which microscope is more powerful, one that has a scale of 100:1 or one that has a scale of 1000:1? Explain. How did you determine the power of the microscope? How do you compare centimeters to millimeters? How do you compare inches to feet? How did you determine the size of the original poster? 7. A different microscope has a scale of 1000 : 1. An amoeba has a length of 25 millimeters under the microscope. What is the actual length of the amoeba? Show your work mm The actual length of the amoeba is millimeter. 8. A centimeter-long paramecium appears to be 17.5 millimeters long under a microscope. What is the power of the microscope? Show your work mm cm With like units, this can be rewritten as 17.5 mm mm The microscope magnifies all images to 50 times their actual size. 9. The height of a building in an architectural drawing is 12 inches. The actual height of the building is 360 feet. What is the scale of the drawing? Show your work. The scale of the drawing is 12 inches : 360 feet. With like units, the scale of the drawing would be 1 foot : 360 feet. The scale would be 1 : A poster was enlarged and made into a billboard. The billboard was 20.5 feet by 36 feet. The scale used was 5 : 1. What was the size of the original poster? Explain your reasoning. The original poster was 4.1 feet by 7.2 feet. The way I determined the original poster size was that I took the dimensions of the poster and divided by 5. So, feet, and feet. 11. How do you determine the scale if a statue is 60 feet high and its scale drawing shows the height as 1 foot high? The scale is written as a ratio of the size of the scale model to the size of the original object. In this case, the scale would be 1 : 60 or Exploring Scale Drawings 461

50 Share Phase, Questions 12 and 13 If the scale on a map was expressed in kilometers, how would you determine the number of miles between two locations? If the scale on a map was expressed in miles, how would you determine the number of kilometers between two locations? How did you decide the scale for the drawing of your math classroom? Did everyone use the same scale? How were the scales different? Can different scales be used to draw your math classroom? 12. Explain how to calculate the actual distance between two cities if you know the distance between them on a map and the scale of the map. First I would take the scale and set up a proportion comparing the scale to the distance measured. Then, I would solve the proportion to come up with the actual distance. For example if the scale was 1 cm : 4 mi and the distance measured was 4 cm my work would look like this: 1 cm 4 cm : 4 mi x mi x = 16 mi 13. Draw a scale drawing of your math classroom. Give the dimensions of the room and the scale. Answers will vary. Remember, you will need to determine the actual size of the room before you can draw it to scale. 462 Chapter 8 Scale Drawings and Scale Factor

51 Problem 5 An example of a blueprint is provided. Students are given certain specifications and they will create a design for a school courtyard using the blueprint and a scale of 1 inch 5 1 foot. 8 Materials Ruler Problem 5 Blueprints A blueprint is a technical drawing, usually of an architectural or engineering design. An example of a blueprint is shown SCALE As students are reading the requirements for Problem 5, they should make note of any questions and highlight or underline important information. Ask pairs of students to summarize the task and to answer any questions. Monitor the discussion to ensure understanding of the material read. 1/8 = Design a courtyard for your school using this blueprint and the scale 1 inch 5 1 foot. 8 Include: features appropriate for a courtyard that would enhance the environment features that would be popular for students, teachers, and parents at least 10 features in the space provided (multiples of the same feature are acceptable) All features should be: drawn to scale positioned on the blueprint keeping scale in mind drawn directly on the blueprint or cut out of paper and taped to the blueprint labeled, either directly on the item or by using a key Grouping Have students complete the blueprint with a partner. Then share the responses as a class. Share Phase, Problem 5 What is a key? What is usually included in a key? What are some of the features you used? Did you include your features in a key? Be prepared to share your solutions and methods. Did your classmates use different features? How can you tell if your blueprint is accurate? Would it have been easier to draw the blueprint if the scale was written in a metric unit? Why or why not? 8.3 Exploring Scale Drawings 463

52 Follow Up Assignment Use the Assignment for Lesson 8.3 in the Student Assignments book. See the Teacher s Resources and Assessments book for answers. Skills Practice Refer to the Skills Practice worksheet for Lesson 8.3 in the Student Assignments book for additional resources. See the Teacher s Resources and Assessments book for answers. Assessment See the Assessments provided in the Teacher s Resources and Assessments book for Chapter 8. Check for Students Understanding The nine planets in the Solar System are Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto. Is this drawing of the nine planets and the Sun drawn to scale? Explain your reasoning. Sun The small planets: Mercury, Venus, Earth, Mars each have diameters less than 13,000 km. The giant planets: Jupiter, Saturn, Uranus, and Neptune each have diameters greater than 48,000 km. This drawing cannot be drawn to scale. It does not appear to have four planets that are significantly larger than the remaining five planets. And the Sun is larger than all of the nine planets and it is not drawn as such. 464 Chapter 8 Scale Drawings and Scale Factor

53 Houses for Our Feathered Friends! Creating Blueprints Learning Goals In this lesson, you will: Use scale drawings to create three-dimensional models.. Use three-dimensional models to create blueprints. Essential Ideas Scale drawings are representations of real objects or places that are in proportion to the real objects or places they represent. The scale is given as a ratio. The scale of a drawing 5 drawing length : actual length The scale of a map 5 map distance : actual distance Texas Essential Knowledge and Skills for Mathematics Grade 7 (5) Proportionality. The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships. The student is expected to: (C) solve mathematical and real-world problems involving similar shape and scale drawings Materials Ruler 8.4 Creating Blueprints 465A

54 Overview Blueprints are the focal point of this lesson. Students encounter a partial blueprint for a square wren house and choose an appropriate unit of measure to complete the blueprint. Finally, students will design their own blueprint for a bird hotel. 465B Chapter 8 Scale Drawings and Scale Factor

55 Warm Up A lighthouse and the blueprint of the lighthouse are shown. 1. What are some features of the lighthouse that are apparent by looking only at the blueprint? Answers will vary. The number of staircases and the number of rest stops can only be seen in the blueprint. 2. What are some features of the lighthouse that are important but not included on the blueprint? Answers will vary. Measurements to indicate the scale of the drawing are not included. It is not possible to know the size of the light house. It could be a miniature lighthouse or a real light house. 8.4 Creating Blueprints 465C

56 465D Chapter 8 Scale Drawings and Scale Factor

57 Houses for Our Feathered Friends! Creating Blueprints Learning Goals In this lesson, you will: Use scale drawings to create three-dimensional models. Use three-dimensional models to create blueprints. The swallows of San Juan Capistrano are famous. They leave Argentina at about the end of October and arrive at the same church every year in California on March 19. How far do these birds travel to their summer vacations? Not far. Just 6000 miles! Do you think there are other creatures that travel long distances at different times of years? Do you think there are any other reasons animals would migrate from one part of the world to another? 8.4 Creating Blueprints 465

58 Problem 1 Students design a blueprint for a square wren house. Problem 1 Rectangular Wren Houses Materials Ruler Grouping Have students complete Questions 1 through 3 with a partner. Then share the responses as a class. Wren houses are built in several sizes and shapes. One example of a square wren house is shown. 1. Label the boards with appropriate measures. Answers will vary. WREN HOUSE A B D E 5 5 1/2 7 A B 6 1/2 6 6 D C E /2 5 1/2 C ALL MATERIAL IS 1/2" THICK BACK PIECE MAY BE ATTACHED WITH 1" SCREWS TO ALLOW FOR TAKING APART FOR CLEANING 2 PIECES 1" DIA HOLE 2. One example of a rectangular wren house is shown. Think about how tall and wide you want the birdhouse to be. Draw the different boards used for this wren house. Include measurements. Answers will vary. 466 Chapter 8 Scale Drawings and Scale Factor

59 Share Phase, Questions 1 through 3 How does seeing the picture of the square wren house help you to draw a blueprint? What are the shapes of the two pieces that compose the roof? What are the shapes of the two pieces that compose the bottom of the birdhouse? What are the shapes of the two pieces that compose the front and the back of the birdhouse? Which parts of the birdhouse are actually squares? How many total pieces of wood are used to build the birdhouse? What scale was used to draw the blueprint of the birdhouse? How much wood would you need to build the birdhouse? 3. You can construct a birdhouse using only nails and a single 1 ft by 6 ft board. Some of the measurements were not included. Label the boards and determine the unknown measurements. (The front and back are made from two pieces.) Answers will vary. Roof C Roof D 4" Floor Side G Side H } chimney Scrap 11 " Back E 11 Front A " 1 5 " 2 Scrap 1 1 " 2 3" 1 9 " " 2 5 " 4 " 4" 1 5 " " 2 Scrap Front B 11 " 11 " Back F F 1 5 " " Creating Blueprints 467

60 Problem 2 Students design a blueprint for a bird hotel. Problem 2 Design Your Own Bird Hotel! Materials Ruler Draw a scale model of a bird hotel. The hotel should have several rooms and separate openings such that each bird can enter its own room. Create a blueprint that includes the measurements necessary to build the birdhouse and include the scale used to draw the model. You may be able to search the Internet for ideas. Answers will vary. Display the following terms for students: scale factor, scale model and scale drawing. Have pairs of students complete the following statements using each vocabulary word: It is important to remember. Scale is different from scale because. Invite students to share their responses with one another. Grouping Have students complete the blueprint with a partner. Then share the responses as a class. Share Phase, Problem 2 What are the shapes of the two pieces that compose the roof for your bird hotel? What is the shape that composes the bottom of your bird hotel? Be prepared to share your solutions and methods. What are the shapes of the two pieces that compose the front and the back of your bird hotel? Which parts of the bird hotel are actually rectangles? How many total pieces of wood are used to build your bird hotel? What scale did you use to draw your blueprint of the bird hotel? How much wood will you need to build your birdhouse? 468 Chapter 8 Scale Drawings and Scale Factor

61 Follow Up Assignment Use the Assignment for Lesson 8.4 in the Student Assignments book. See the Teacher s Resources and Assessments book for answers. Skills Practice Refer to the Skills Practice worksheet for Lesson 8.4 in the Student Assignments book for additional resources. See the Teacher s Resources and Assessments book for answers. Assessment See the Assessments provided in the Teacher s Resources and Assessments book for Chapter 8. Check for Students Understanding Jeff started to draw a blueprint of his dream house but was interrupted. He wanted to show this blueprint to his family. Add what you think might be important information to the blueprint that would be of interest to his family. Answers will vary. Students could designate dimensions of the rooms, the names of the rooms, locations of steps, windows, and doorways. 8.4 Creating Blueprints 468A

62 468B Chapter 8 Scale Drawings and Scale Factor

63 Chapter 8 Summary Key Terms scale factor (8.1) aspect ratio (8.2) scale drawings (8.3) Dilating Scale Drawings Scale drawings are used to display very large or very small objects. Maps and blueprints are examples of scale drawings. The ratio of lengths in an enlargement to those of the original figure is called the scale factor. One way to dilate, or enlarge or shrink, a scale drawing is to use a grid. Example The drawing of the car is enlarged on the grid. Did you like drawing these scale figures or do you like building things with your hands? If you do, the part of your brain that controls your fingers is much bigger than those who don, t work with their hands! Chapter 8 Summary 469